Sutra ću joj čestitati rođendan porukom.

A natural breakdown is:

• Sutra = tomorrow
• ću = I will
• joj = to her / her
• čestitati = to congratulate / to wish someone well for an occasion
• rođendan = birthday
• porukom = by message, with a message, via message

So the sentence structure is roughly:

• Tomorrow + will + to her + congratulate/wish + birthday + by message

Natural English would be something like Tomorrow I’ll wish her a happy birthday by message.

Because Croatian often drops subject pronouns when the verb already shows who the subject is.

Here, ću already tells you it is first person singular: I will.

So:

• Sutra ću joj čestitati rođendan porukom = normal
• Ja ću joj čestitati rođendan porukom = also possible, but ja adds emphasis, like I will do it

English usually needs the subject pronoun, but Croatian often does not.

This is the future I tense in Croatian:

• ću = future auxiliary for I
• čestitati = infinitive

So:

• ću čestitati = I will congratulate / I will wish

The full pattern is basically:

• subject (optional) + future auxiliary + infinitive

In this sentence, the subject is omitted because it is already clear from ću.

Yes. Both patterns exist.

In your sentence, Sutra comes first, and Croatian clitics like ću usually go in the second position of the clause:

• Sutra ću joj čestitati rođendan porukom.

You can also put the infinitive first and then attach the future auxiliary after it:

• Čestitat ću joj rođendan sutra porukom.

With verbs like čestitati, this often becomes:

• čestitat ću rather than čestitati ću

So both are correct, but the sentence you were given follows the very common second-position clitic pattern.

Because Croatian clitics have a fairly fixed internal order.

Here both ću and joj are clitics, and the normal order is:

• ću first
• joj after it

So:

• Sutra ću joj čestitati... = standard

Not:

• Sutra joj ću čestitati... = not standard

This is something learners usually just have to get used to as part of Croatian clitic order.

Joj is dative singular of ona.

It is used because čestitati normally takes:

• the person in the dative
• the occasion/thing in the accusative

So the pattern is:

• čestitati nekome nešto
  = to congratulate someone on something

That is why:

• joj = to her
• nju would be wrong here, because nju is accusative, not dative

No preposition is needed with čestitati in this structure.

Croatian says:

• čestitati nekome rođendan

So rođendan is the thing being congratulated/wished, and it is in the accusative.

The important detail is that rođendan is a masculine inanimate noun, and in the singular its accusative looks the same as its nominative:

• nominative: rođendan
• accusative: rođendan

So even though the case is accusative, the form does not change.

Because Croatian does not need it here.

In English, you normally say her birthday, but Croatian can express that idea through joj:

• čestitati joj rođendan already means you are congratulating her on her birthday

So adding njezin or njen is usually unnecessary:

• čestitati joj njezin rođendan = understandable, but heavier and less natural in many contexts

Croatian often leaves out possessives when the meaning is already clear from the pronoun or context.

It can cover both ideas, depending on context.

With rođendan, English usually sounds more natural with:

• wish her a happy birthday

But Croatian uses čestitati very naturally here.

So:

• čestitati joj rođendan = wish her a happy birthday / congratulate her on her birthday

You can also make it more explicit:

• čestitati joj sretan rođendan

But the shorter version in your sentence is completely normal.

Porukom is the instrumental singular of poruka.

The instrumental case is often used to show the means or method by which something is done.

So:

• porukom = by message, with a message, via message

This is similar to expressions like:

• autobusom = by bus
• telefonom = by phone
• mailom = by email

So in this sentence, porukom tells you how the birthday wish will be sent.

Not necessarily. Poruka is a general word for message.

Depending on context, porukom could mean:

• a text message
• a chat message
• some other written message

In modern everyday use, people often understand it as a phone or chat message if that makes sense from context.

If you want to be more specific, you could say things like:

• SMS-om = by SMS
• tekstualnom porukom = by text message
• u poruci = in a message

But porukom on its own is perfectly natural.

Yes, Croatian word order is fairly flexible, but not completely free.

Because cases show grammatical roles, you can move parts around for emphasis:

• Sutra ću joj čestitati rođendan porukom.
• Sutra ću joj porukom čestitati rođendan.
• Rođendan ću joj sutra čestitati porukom.

These are not all equally neutral, though.

The most important restriction is that clitics like ću and joj tend to stay near the beginning, in the normal clitic position.

So the original sentence is natural, and Sutra ću joj porukom čestitati rođendan is also very natural.

Not in the same way.

With čestitati, Croatian normally uses:

• čestitati nekome rođendan
• čestitati nekome sretan rođendan

Using za rođendan would usually express a different idea, more like for a birthday in the sense of purpose or occasion:

• Kupio sam joj poklon za rođendan. = I bought her a present for her birthday.

So with čestitati, it is better to keep the direct pattern used in your sentence.